Optimal. Leaf size=1645 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.75844, antiderivative size = 1657, normalized size of antiderivative = 1.01, number of steps used = 47, number of rules used = 15, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.484, Rules used = {2498, 2513, 2411, 43, 2334, 12, 14, 2301, 2418, 2389, 2295, 2394, 2393, 2391, 2395} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2498
Rule 2513
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rule 2418
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rubi steps
\begin{align*} \int (g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{(2 b p r) \int \frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx}{3 h}-\frac{(2 d q r) \int \frac{(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{3 h}\\ &=\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{\left (2 b p^2 r^2\right ) \int \frac{(g+h x)^3 \log (a+b x)}{a+b x} \, dx}{3 h}-\frac{\left (2 b p q r^2\right ) \int \frac{(g+h x)^3 \log (c+d x)}{a+b x} \, dx}{3 h}-\frac{\left (2 d p q r^2\right ) \int \frac{(g+h x)^3 \log (a+b x)}{c+d x} \, dx}{3 h}-\frac{\left (2 d q^2 r^2\right ) \int \frac{(g+h x)^3 \log (c+d x)}{c+d x} \, dx}{3 h}+\frac{\left (2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{(g+h x)^3}{a+b x} \, dx}{3 h}+\frac{\left (2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{(g+h x)^3}{c+d x} \, dx}{3 h}\\ &=\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{\left (2 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{b g-a h}{b}+\frac{h x}{b}\right )^3 \log (x)}{x} \, dx,x,a+b x\right )}{3 h}-\frac{\left (2 b p q r^2\right ) \int \left (\frac{h (b g-a h)^2 \log (c+d x)}{b^3}+\frac{(b g-a h)^3 \log (c+d x)}{b^3 (a+b x)}+\frac{h (b g-a h) (g+h x) \log (c+d x)}{b^2}+\frac{h (g+h x)^2 \log (c+d x)}{b}\right ) \, dx}{3 h}-\frac{\left (2 d p q r^2\right ) \int \left (\frac{h (d g-c h)^2 \log (a+b x)}{d^3}+\frac{(d g-c h)^3 \log (a+b x)}{d^3 (c+d x)}+\frac{h (d g-c h) (g+h x) \log (a+b x)}{d^2}+\frac{h (g+h x)^2 \log (a+b x)}{d}\right ) \, dx}{3 h}-\frac{\left (2 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{d g-c h}{d}+\frac{h x}{d}\right )^3 \log (x)}{x} \, dx,x,c+d x\right )}{3 h}+\frac{\left (2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \left (\frac{h (b g-a h)^2}{b^3}+\frac{(b g-a h)^3}{b^3 (a+b x)}+\frac{h (b g-a h) (g+h x)}{b^2}+\frac{h (g+h x)^2}{b}\right ) \, dx}{3 h}+\frac{\left (2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \left (\frac{h (d g-c h)^2}{d^3}+\frac{(d g-c h)^3}{d^3 (c+d x)}+\frac{h (d g-c h) (g+h x)}{d^2}+\frac{h (g+h x)^2}{d}\right ) \, dx}{3 h}\\ &=-\frac{p^2 r^2 \log (a+b x) \left (\frac{18 h (b g-a h)^2 (a+b x)}{b^3}+\frac{9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac{2 h^3 (a+b x)^3}{b^3}+\frac{6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{18 h (d g-c h)^2 (c+d x)}{d^3}+\frac{9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac{2 h^3 (c+d x)^3}{d^3}+\frac{6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}+\frac{2 (b g-a h)^2 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^2}+\frac{2 (d g-c h)^2 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^2}+\frac{(b g-a h) p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b h}+\frac{(d g-c h) q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d h}+\frac{2 p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 (b g-a h)^3 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^3 h}+\frac{2 (d g-c h)^3 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^3 h}+\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac{\left (2 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{h x \left (18 b^2 g^2+9 b g h (-4 a+x)+h^2 \left (18 a^2-9 a x+2 x^2\right )\right )+6 (b g-a h)^3 \log (x)}{6 b^3 x} \, dx,x,a+b x\right )}{3 h}-\frac{1}{3} \left (2 p q r^2\right ) \int (g+h x)^2 \log (a+b x) \, dx-\frac{1}{3} \left (2 p q r^2\right ) \int (g+h x)^2 \log (c+d x) \, dx-\frac{\left (2 (b g-a h) p q r^2\right ) \int (g+h x) \log (c+d x) \, dx}{3 b}-\frac{\left (2 (b g-a h)^2 p q r^2\right ) \int \log (c+d x) \, dx}{3 b^2}-\frac{\left (2 (b g-a h)^3 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b^2 h}-\frac{\left (2 (d g-c h) p q r^2\right ) \int (g+h x) \log (a+b x) \, dx}{3 d}-\frac{\left (2 (d g-c h)^2 p q r^2\right ) \int \log (a+b x) \, dx}{3 d^2}-\frac{\left (2 (d g-c h)^3 p q r^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 d^2 h}+\frac{\left (2 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{h x \left (18 d^2 g^2+9 d g h (-4 c+x)+h^2 \left (18 c^2-9 c x+2 x^2\right )\right )+6 (d g-c h)^3 \log (x)}{6 d^3 x} \, dx,x,c+d x\right )}{3 h}\\ &=-\frac{(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac{2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{18 h (b g-a h)^2 (a+b x)}{b^3}+\frac{9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac{2 h^3 (a+b x)^3}{b^3}+\frac{6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}-\frac{(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac{2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac{2 (b g-a h)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{18 h (d g-c h)^2 (c+d x)}{d^3}+\frac{9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac{2 h^3 (c+d x)^3}{d^3}+\frac{6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac{2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac{2 (b g-a h)^2 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^2}+\frac{2 (d g-c h)^2 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^2}+\frac{(b g-a h) p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b h}+\frac{(d g-c h) q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d h}+\frac{2 p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 (b g-a h)^3 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^3 h}+\frac{2 (d g-c h)^3 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^3 h}+\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac{\left (p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{h x \left (18 b^2 g^2+9 b g h (-4 a+x)+h^2 \left (18 a^2-9 a x+2 x^2\right )\right )+6 (b g-a h)^3 \log (x)}{x} \, dx,x,a+b x\right )}{9 b^3 h}+\frac{\left (2 b p q r^2\right ) \int \frac{(g+h x)^3}{a+b x} \, dx}{9 h}+\frac{\left (2 d p q r^2\right ) \int \frac{(g+h x)^3}{c+d x} \, dx}{9 h}+\frac{\left (d (b g-a h) p q r^2\right ) \int \frac{(g+h x)^2}{c+d x} \, dx}{3 b h}-\frac{\left (2 (b g-a h)^2 p q r^2\right ) \operatorname{Subst}(\int \log (x) \, dx,x,c+d x)}{3 b^2 d}+\frac{\left (2 d (b g-a h)^3 p q r^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 h}+\frac{\left (b (d g-c h) p q r^2\right ) \int \frac{(g+h x)^2}{a+b x} \, dx}{3 d h}-\frac{\left (2 (d g-c h)^2 p q r^2\right ) \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{3 b d^2}+\frac{\left (2 b (d g-c h)^3 p q r^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 d^3 h}+\frac{\left (q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{h x \left (18 d^2 g^2+9 d g h (-4 c+x)+h^2 \left (18 c^2-9 c x+2 x^2\right )\right )+6 (d g-c h)^3 \log (x)}{x} \, dx,x,c+d x\right )}{9 d^3 h}\\ &=\frac{2 (b g-a h)^2 p q r^2 x}{3 b^2}+\frac{2 (d g-c h)^2 p q r^2 x}{3 d^2}-\frac{2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac{(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac{2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{18 h (b g-a h)^2 (a+b x)}{b^3}+\frac{9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac{2 h^3 (a+b x)^3}{b^3}+\frac{6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}-\frac{2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac{(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac{2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac{2 (b g-a h)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{18 h (d g-c h)^2 (c+d x)}{d^3}+\frac{9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac{2 h^3 (c+d x)^3}{d^3}+\frac{6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac{2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac{2 (b g-a h)^2 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^2}+\frac{2 (d g-c h)^2 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^2}+\frac{(b g-a h) p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b h}+\frac{(d g-c h) q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d h}+\frac{2 p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 (b g-a h)^3 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^3 h}+\frac{2 (d g-c h)^3 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^3 h}+\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac{\left (p^2 r^2\right ) \operatorname{Subst}\left (\int \left (h \left (18 (b g-a h)^2+9 h (b g-a h) x+2 h^2 x^2\right )+\frac{6 (b g-a h)^3 \log (x)}{x}\right ) \, dx,x,a+b x\right )}{9 b^3 h}+\frac{\left (2 b p q r^2\right ) \int \left (\frac{h (b g-a h)^2}{b^3}+\frac{(b g-a h)^3}{b^3 (a+b x)}+\frac{h (b g-a h) (g+h x)}{b^2}+\frac{h (g+h x)^2}{b}\right ) \, dx}{9 h}+\frac{\left (2 d p q r^2\right ) \int \left (\frac{h (d g-c h)^2}{d^3}+\frac{(d g-c h)^3}{d^3 (c+d x)}+\frac{h (d g-c h) (g+h x)}{d^2}+\frac{h (g+h x)^2}{d}\right ) \, dx}{9 h}+\frac{\left (d (b g-a h) p q r^2\right ) \int \left (\frac{h (d g-c h)}{d^2}+\frac{(d g-c h)^2}{d^2 (c+d x)}+\frac{h (g+h x)}{d}\right ) \, dx}{3 b h}+\frac{\left (2 (b g-a h)^3 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 h}+\frac{\left (b (d g-c h) p q r^2\right ) \int \left (\frac{h (b g-a h)}{b^2}+\frac{(b g-a h)^2}{b^2 (a+b x)}+\frac{h (g+h x)}{b}\right ) \, dx}{3 d h}+\frac{\left (2 (d g-c h)^3 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 d^3 h}+\frac{\left (q^2 r^2\right ) \operatorname{Subst}\left (\int \left (h \left (18 (d g-c h)^2+9 h (d g-c h) x+2 h^2 x^2\right )+\frac{6 (d g-c h)^3 \log (x)}{x}\right ) \, dx,x,c+d x\right )}{9 d^3 h}\\ &=\frac{8 (b g-a h)^2 p q r^2 x}{9 b^2}+\frac{2 (b g-a h) (d g-c h) p q r^2 x}{3 b d}+\frac{8 (d g-c h)^2 p q r^2 x}{9 d^2}+\frac{5 (b g-a h) p q r^2 (g+h x)^2}{18 b h}+\frac{5 (d g-c h) p q r^2 (g+h x)^2}{18 d h}+\frac{4 p q r^2 (g+h x)^3}{27 h}+\frac{2 (b g-a h)^3 p q r^2 \log (a+b x)}{9 b^3 h}+\frac{(b g-a h)^2 (d g-c h) p q r^2 \log (a+b x)}{3 b^2 d h}-\frac{2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac{(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac{2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{18 h (b g-a h)^2 (a+b x)}{b^3}+\frac{9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac{2 h^3 (a+b x)^3}{b^3}+\frac{6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}+\frac{(b g-a h) (d g-c h)^2 p q r^2 \log (c+d x)}{3 b d^2 h}+\frac{2 (d g-c h)^3 p q r^2 \log (c+d x)}{9 d^3 h}-\frac{2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac{(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac{2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac{2 (b g-a h)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{18 h (d g-c h)^2 (c+d x)}{d^3}+\frac{9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac{2 h^3 (c+d x)^3}{d^3}+\frac{6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac{2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac{2 (b g-a h)^2 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^2}+\frac{2 (d g-c h)^2 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^2}+\frac{(b g-a h) p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b h}+\frac{(d g-c h) q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d h}+\frac{2 p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 (b g-a h)^3 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^3 h}+\frac{2 (d g-c h)^3 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^3 h}+\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{2 (d g-c h)^3 p q r^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 d^3 h}-\frac{2 (b g-a h)^3 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 h}+\frac{\left (p^2 r^2\right ) \operatorname{Subst}\left (\int \left (18 (b g-a h)^2+9 h (b g-a h) x+2 h^2 x^2\right ) \, dx,x,a+b x\right )}{9 b^3}+\frac{\left (2 (b g-a h)^3 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 h}+\frac{\left (q^2 r^2\right ) \operatorname{Subst}\left (\int \left (18 (d g-c h)^2+9 h (d g-c h) x+2 h^2 x^2\right ) \, dx,x,c+d x\right )}{9 d^3}+\frac{\left (2 (d g-c h)^3 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 d^3 h}\\ &=\frac{2 (b g-a h)^2 p^2 r^2 x}{b^2}+\frac{8 (b g-a h)^2 p q r^2 x}{9 b^2}+\frac{2 (b g-a h) (d g-c h) p q r^2 x}{3 b d}+\frac{8 (d g-c h)^2 p q r^2 x}{9 d^2}+\frac{2 (d g-c h)^2 q^2 r^2 x}{d^2}+\frac{h (b g-a h) p^2 r^2 (a+b x)^2}{2 b^3}+\frac{2 h^2 p^2 r^2 (a+b x)^3}{27 b^3}+\frac{h (d g-c h) q^2 r^2 (c+d x)^2}{2 d^3}+\frac{2 h^2 q^2 r^2 (c+d x)^3}{27 d^3}+\frac{5 (b g-a h) p q r^2 (g+h x)^2}{18 b h}+\frac{5 (d g-c h) p q r^2 (g+h x)^2}{18 d h}+\frac{4 p q r^2 (g+h x)^3}{27 h}+\frac{2 (b g-a h)^3 p q r^2 \log (a+b x)}{9 b^3 h}+\frac{(b g-a h)^2 (d g-c h) p q r^2 \log (a+b x)}{3 b^2 d h}-\frac{2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac{(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac{2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}+\frac{(b g-a h)^3 p^2 r^2 \log ^2(a+b x)}{3 b^3 h}-\frac{p^2 r^2 \log (a+b x) \left (\frac{18 h (b g-a h)^2 (a+b x)}{b^3}+\frac{9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac{2 h^3 (a+b x)^3}{b^3}+\frac{6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}+\frac{(b g-a h) (d g-c h)^2 p q r^2 \log (c+d x)}{3 b d^2 h}+\frac{2 (d g-c h)^3 p q r^2 \log (c+d x)}{9 d^3 h}-\frac{2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac{(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac{2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac{2 (b g-a h)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}+\frac{(d g-c h)^3 q^2 r^2 \log ^2(c+d x)}{3 d^3 h}-\frac{q^2 r^2 \log (c+d x) \left (\frac{18 h (d g-c h)^2 (c+d x)}{d^3}+\frac{9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac{2 h^3 (c+d x)^3}{d^3}+\frac{6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac{2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac{2 (b g-a h)^2 p r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^2}+\frac{2 (d g-c h)^2 q r x \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^2}+\frac{(b g-a h) p r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b h}+\frac{(d g-c h) q r (g+h x)^2 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d h}+\frac{2 p r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 q r (g+h x)^3 \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{9 h}+\frac{2 (b g-a h)^3 p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 b^3 h}+\frac{2 (d g-c h)^3 q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{3 d^3 h}+\frac{(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac{2 (d g-c h)^3 p q r^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 d^3 h}-\frac{2 (b g-a h)^3 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 h}\\ \end{align*}
Mathematica [A] time = 1.89012, size = 899, normalized size = 0.55 \[ \frac{-18 a \left (3 b^2 g^2-3 a b h g+a^2 h^2\right ) p^2 r^2 \log ^2(a+b x) d^3-6 p r \log (a+b x) \left (6 c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) q r \log (c+d x) b^3-6 (b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2+a d h (c h-3 d g) b+a^2 d^2 h^2\right ) q r \log \left (\frac{b (c+d x)}{b c-a d}\right )+a d \left (\left (6 \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) q b^2-3 a d h (3 d g (3 p+q)-c h q) b+a^2 d^2 h^2 (11 p+2 q)\right ) r-6 d^2 \left (3 b^2 g^2-3 a b h g+a^2 h^2\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+b \left (-18 b^2 c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) q^2 r^2 \log ^2(c+d x)-6 q r \left (\left (c \left (18 d^2 (p+q) g^2-9 c d h (p+3 q) g+c^2 h^2 (2 p+11 q)\right ) b^2-3 a d \left (6 d^2 g^2+6 c d h g-c^2 h^2\right ) p b+6 a^2 c d^2 h^2 p\right ) r-6 b^2 c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (c+d x)+d \left (\left (x \left (6 c^2 q (8 p+11 q) h^2-3 c d q (p+q) (54 g+5 h x) h+d^2 (p+q)^2 \left (108 g^2+27 h x g+4 h^2 x^2\right )\right ) b^2-3 a p \left (\left (-36 q g^2+54 h (p+q) x g+5 h^2 (p+q) x^2\right ) d^2-12 c h q (h x-3 g) d-12 c^2 h^2 q\right ) b+6 a^2 d^2 h^2 p (11 p+8 q) x\right ) r^2-6 \left (x \left ((p+q) \left (18 g^2+9 h x g+2 h^2 x^2\right ) d^2-3 c h q (6 g+h x) d+6 c^2 h^2 q\right ) b^2+3 a d^2 p \left (6 g^2-6 h x g-h^2 x^2\right ) b+6 a^2 d^2 h^2 p x\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) r+18 b^2 d^2 x \left (3 g^2+3 h x g+h^2 x^2\right ) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+36 (b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2+a d h (c h-3 d g) b+a^2 d^2 h^2\right ) p q r^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{54 b^3 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.385, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{2} \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37896, size = 1516, normalized size = 0.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (h^{2} x^{2} + 2 \, g h x + g^{2}\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (h x + g\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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